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2018 Houston Summer School on Dynamical Systems

# Houston Summer School on Dynamical Systems

## Department of MathematicsUniversity of Houston

The Department of Mathematics at University of Houston will host the sixth annual Houston Summer School on Dynamical Systems from May 16-24, 2018.

As in past years, the school is designed for graduate students; however, there will also be an opportunity for undergraduate students to participate, arriving two days early and then staying for the main event.  See below for details.

The school will use short lecture courses, tutorial and discussion sessions, and student projects to explore various topics in dynamical systems.   It will be accessible to students without a background in dynamics, but is also intended for students who have begun studying dynamics and wish to learn more about this field.

We anticipate being able to provide lodging for all participants, and to reimburse travel expenses.

### School Poster

A number of participant spots are reserved for undergraduate students, who will first attend several preliminary lectures on May 14-15 to introduce concepts needed for the short courses that may not have appeared in the students' undergraduate coursework so far.  There will also be problem sessions, discussion, and Q&A time on those days, as well as continuing review sessions during the school itself that are specifically targeted at the undergraduate participants, with the goal of helping them follow the graduate-level material being presented.

Undergraduate students interested in participating in the event should apply following the instructions below.

## Descriptions of short courses

The following short courses are planned:
• Basics of ergodic theory
(Alan Haynes, Joanna Furno - University of Houston)
These lectures are intended to bring all participants up to speed with basic definitions, results, and proofs from the theory of measure preserving dynamical systems, including: recurrence, the Birkhoff ergodic theorem, spaces of measures, Koopman operators and spectral theory, basic entropy theory, and techniques for studying dynamics on compact groups.

• Statistical properties in hyperbolic dynamics
(Matthew Nicol, William Ott, Andrew Török - University of Houston)
These lectures will introduce the notion of decay of correlations for a dynamical system and will describe two important methods for establishing a rate of decay: spectral gap (Perron-Frobenius theory) and Birkhoff cones.

• Statistical mechanics and thermodynamic formalism
(Vaughn Climenhaga - University of Houston)
These lectures will introduce basic models for statistical mechanics of lattice gases and explain how the mathematical techniques used in their analysis can be translated to the setting of dynamical systems, leading to the theory of thermodynamic formalism, equilibrium states, and SRB measures.

• Dynamics of quantum spin systems
(Anna Vershynina - University of Houston)
These lectures will present the analysis of dynamics in quantum systems. The main question of consideration here is how fast does information spread in quantum systems? The direct answer to this question could be "instantaneously", due to a quantum phenomena called entanglement. Entanglement allows far away quantum particles to "feel" each instantaneously in a way that is stronger than any classical correlation. Fortunately for the future of the quantum computer, this answer is not the end of the story. The ground-breaking discovery is that, up to exponentially-small error, the information spreads with finite speed even in quantum systems, making them essentially robust against small perturbations. This is known as Lieb-Robinson bounds, or locality estimates.

• Dynamical approaches to the spectral theory of operators
(David Damanik - Rice University)
The goal of these lectures is to give an overview of foundational techniques and results underlying dynamical approaches to the study of spectra of operators associated to physical systems, such as Schrödinger operators associated to graphs. Topics to be covered include cocycles, Lyapunov exponents, rotation numbers, uniform hyperbolicity, and reducibility. A major theme will be the study of systems with potentials displaying "aperiodic order," i.e. between random and periodic, and a number of examples, such as the Fibonacci Hamiltonian, will be considered in detail.

• Dynamics on homogeneous spaces, with applications to number theory
(Seonhee Lim - Seoul National University)
The lectures will begin with the introduction of closed linear groups, lattices, and natural dynamical systems on their quotients. Motivating examples will be actions defined by multiplication by diagonal matrices on quotients of $$\mathrm{PSL}_n(\mathbb{R})$$. After explaining Hopf's proof of the ergodicity of the geodesic flow, these lectures will aim to provide an overview of the proof by Margulis of the Oppenheim conjecture. If time permits they will also include a discussion of Eskin, Margulis, and Mozes's proof(s) of quantitative versions of the Oppenheim conjecture.

## Prerequisites and application process

Graduate students participating in the school should be familiar with the following prerequisite material: measure theory, basic functional analysis, and algebra (mainly concerning groups and linear algebra). There will be review sessions during the school to give a brief overview of the most relevant parts of these topics.

To apply for participation in the summer school please complete this form with the following information:
1. Your name, current institution, and program and year of study.  Please also include the name and email address of your Ph.D. advisor or of another mathematician who can serve as a reference if necessary.
2. A list of recent mathematics courses you have taken and the grades earned.  Please indicate your background in the prerequisite topics of measure theory, functional analysis, and algebra (mainly concerning groups and linear algebra).
3. A brief description of your mathematical interests, particularly as they relate to the topic of the summer school.

Undergraduate students interested in participating should follow the instructions above, and should also arrange to have a professor send a brief letter of recommendation to Vaughn Climenhaga at climenha@math.uh.edu. This letter need not be long, but should attest to the student's overall level of preparation and ability to quickly grasp the essential parts of advanced topics, which will be important in order to follow the short courses at the school.

To contact the organizers, please use uh.summer.school@gmail.com.
The deadline for applications to be guaranteed full consideration is February 28, 2018.

Funding for this event is provided by the NSF grant DMS-1800669.

[most of these open in the same, separate, tab]

DIRECTIONS to PGH from the bus stop (marked stop is Cullen & Hollman on Metro line 4)

Videos of lectures

Schedule of lectures

Possible evening and weekend activities

Notes for the undergraduate prep sessions

Crash course in topology by Alan Haynes

List of project papers

More about decay of correlations via the transfer operator

Exercises:

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